Gravitational Potential Energy - Video Tutorials & Practice Problems
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1
concept
Gravitational Potential Energy
Video duration:
6m
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Hey guys. So now we've seen how masses exert gravitational forces on each other. We're gonna see how to solve problems involving gravitational potential energy. So we're gonna involve questions involving changing energies and velocities. But we're going to see that we need a new equation for that potential energy to solve these problems. OK. So that's the, so let's think back to a second when we were first introduced to energy, we had one object that was above the ground. All we needed was just MG and H we just ignored that the earth had mass. But now when we're dealing with gravitation, we're always going to have problems that involve multiple masses and distances. So now the new equation that we're gonna use from now on for gravitational potential energy is negative GMM over R. OK. So we have to be very careful when we use this equation because there's two things. One, there's a negative sign here. It kind of looks like Newton's law of gravity with this negative sign. And there's also only one power of R in the denominator. Now, just remember that little R represents the center of mass distance, big R plus H So if you're talking about a planet, you have to go all the way from center to center. OK. Now, in fact, what actually happens is that these equations, this equation becomes this equation here on the left kind of like the simplified version for very small changes in height. So we could use this equation here because we always assume that the heights were very, very small, so very small delta H this equation actually becomes this one. All right. So we used U equals MGH as basically an alternative for when we solved kinematics problems. So let's just say for example, that I had a question involving dropping this space rock from some height. And I wanted to figure out what the velocity was when it hit the ground V final. Now, I had two approaches, I could use kinematics, right. So I could figure out what G is I just needed to know the delta Y or the height I have the V knots. And I could use kinematics to solve these things. The reason I could use kinematics is because we just assumed that this G was 9.8 and we just assumed that it was constant. So we could use our kinematics equation, things like that because the acceleration is constant, we also could have used the conservation of energy. Well, the difference is that in gravitation, the main difference is that we cannot use kinematics to solve these problems. The reason for that is that as these two, these two masses, as they get farther and farther apart, as there are changes, the gravitational forces and the accelerations between them will also change. So the reason we cannot use kinematics is because as R changes G is not constant, it's changing. But the good thing is is that we can still use gravitational potential energy or sorry, we can use energy conservation to solve these problems. The only thing that changes is that instead of just using this MGH for U I and UF, we're just gonna use this new equation negative GMM over R. That's the whole thing. So we're just gonna replace that thing over there. OK. So let's go ahead and take a look at an example. We've got an asteroid that is falling to earth from some distance above the center and we're ask what's gonna be the impact speed. So in other words, we've got this space rock over here, we've got the earth. All right. So we've got these two things. Now, I've got the initial distance that's gonna be over here. And then what happens is that the asteroid falls towards the earth and it's gonna hit right here at the center. So what I'm supposed to find out is I'm supposed to find out what is the, the final. So I know that when it hits the surface, there's still gonna be some R over here. So let's go ahead and set up my energy conservation because I have an initial and I have a final right here. I've got an initial velocity and a final velocity. I definitely need to use K uh energy conservation. I can't use kinematics because the G changes across this distance. So I've got initial uh actually, I can just go ahead and I got it right here. Boom. So I got my energy conservation. First things. First, is anything moving in the initial? And is there any kinetic energy? Well, I've got the space rock is dropped. So it's at rest and then the earth we can just assume is not moving. So there's no kinetic energy over here. Now there is some potential energy because we have two masses, right? We've got the mass of the asteroid, the mass of the earth separated by some distance. So I've got negative G big M little M over our initial. So I've had to use that new gravitational potential energy. Now, is there any work done by non conservative forces? But we're always talking about gravity and gravity is a conservative force. So there's no work done by non conserv forces. Now, finally, I have the kinetic energy final. Well, that just is going to be one half of the earth velocity squared plus one half of the asteroid velocity squared. Now, we can just assume that the earth is so massive that it's not actually moving. Some of the words, this uh term is gonna cancel out it won't always cancel out in some cases. But in this case, it does. And this is actually where our target variable comes from. This is we're gonna get that velocity. And now finally, we have the potential energy. Let's take a look here. Is there some potential energy when it hits the surface? Well, actually there's still some center of mass distance in between the asteroid and the earth. So that means that there is actually still some gravitational potential energy even at the surface. Remember that this R final is gonna be big R plus H. But when you're at the surface, what happens is that your H just turns to zero but your little R just becomes big R. So that means that there's still some gravitational potential energy at the surface. It's just this, the little R turns into big R. So that's what we're gonna put it over here. All right, that's basically it. So now we just have to go ahead and rearrange. I'm just gonna move this term all the way over to the left and it's gonna become positive because of the negative sign. And then I'm gonna have GMM over big R minus GMM over R initial equals one half. And I've got MV final squared. So what I'm gonna do is I'm gonna move this one half over to the other side and I'm also gonna pull out this GMM as the greatest common factor, see how it appears on both sides here. So I've got GMM then I've got one over R oh Sorry. There's a two out here, one over R minus one over R initial equals V final squared. And actually I missed an M. So there's an MV final squared. So this is my target variable here. Let's just go ahead and plug in everything. But there's one last thing if you look through I G I've got the mass and I've got the RS and because all of these things are constants over here, the only thing I'm missing is the mass of the asteroid. But just like what happens in normal energy calculations, this uh this m will actually cancel out because it's, it appears on both sides of the equation. There we go. OK. So now all we have to do is take the square roots. So we've got the final velocity is going to be the square roots. And I'm gonna have 26.67 times 10 to the minus 11. The mass of the earth is 5.97 times 10 to the 24th. And now I've got uh the distances. So I've got 1/6 0.37 times 10 to the sixth minus 1/6 times 10 to the seventh. So you have to be very careful when you plug that all into your calculators. You should get a final velocity 1.06 times 10 to the fourth meters per second. And that's the answer. All right. So that's how you solve these gravitational potential energy problems. Let me know if you guys have any questions.
2
Problem
Problem
How much energy is required to move a 1000-kg object from Earth's surface to a height twice Earth's radius?
A
J
B
J
C
J
D
J
3
example
Collision of Two Small Planets
Video duration:
5m
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Hey guys. So let's check out this problem. We have two identical masses, two identical planets that are initially at rest, but they're eventually going to drift together due to gravity and collide. And we're supposed to be finding what the speed of each of these planets are. So because I'm talking about speed and gravitation, I know I'm gonna need to use some energy. I can't use kinematics because the forces are constantly changing force, gravity constantly changes. So if I'm using energy considerations, I need an initial and a final position. So let's go ahead and draw a diagram. I've got these two masses that are some distance apart and I'm told what the masses of each of those things are. So M equals seven times 10 to the 22. And I'm told that their center of mass distance here is equal to uh which is our initial is equal to two. Sorry, I've got five times 10 to the 10 and that's in meters. So eventually these things are going to crash into each other. So they initially start from rest. So we know the initial velocity is equal to zero. But eventually, when these two things collide with each other, their surfaces are going to touch like that. And we know that the distance between them is going to be this final distance. RF and they're both going to have some final velocity VF. And that's actually what we're looking for, what is the speed of each of these planets as they collide. So that is our target variable here. So we need VF but we need to start with our energy conservation laws. So let's go ahead and do that. I've got the initial and potential of the initial kinetic and potential. Uh plus any work that's done is equal to the final kinetic and potential. So let's see these things are initially starting from rest. So we know that the initial velocities of both of them are zero. What that means is that the kinetic energy initial is going to be zero. We also know that we're talking about gravity and gravity is pulling these things together and it's a conservative force. So that means that there is no non conservative forces acting on this. So let's go ahead and write out each of these uh equations. Now, we know that the initial gravitational potential energy is GM. And the other one is also M these are MS are identical because we're given the mass of both planets and then that's gonna equal to um so divided by the R I and then we've got equal to, let's see what about the kinetic energy final. Well, normally, if we were talking about one object, we would just do one half MV squared. But we're actually talking about this energy conservation law for the entire system. Now remember that this whole entire equation for energy conservation applies to all objects that are in the system. So it's not just 11 half MV squared, we also have one half MV squared for the other object. But because the MS are the same for both of them, I don't have to write M one M two. It's just M and then I've got the gravitational potential. So GMM again, divided by now the final distance. So really what my target variable is in this uh equation are these V finals. So I can go ahead and clean up this equation a little bit. So I'm gonna write this as GM squared because the both of those MS are the same. So I've got GM squared divided by our initial equals. And then I've got one half MV squared plus another one half MV squared. So that's gonna be MV final squared. And then I've got minus GMM divided by R or sorry GM squared. So GM squared divided by R final. But when it reaches this spot here, and these two things are colliding. What is that? What is that final distance between them? The centers of masses? Well, think about it if this is just the radius of one and the radius of another, their surfaces have to be touching. So that means that this final distance here is actually two times the radius, that's the point at which they finally collide. So two times big R, which we actually know. So I'm gonna go ahead and write that big R right here. So now, basically, I've got to isolate this V final squared by moving this term over to the other side. Also, what I can do is I can actually cancel one M factor that appears in each one of these things. So now that I finally cleaned this up, I've got G times M divided by, well, this thing is gonna be positive when it goes over. So it's gonna be a two R and then minus this term over here. So it's gonna be GM. Oops. So I've got GM over R initial equals V final squared. So now, finally, all I can do, uh is I can pull out this GM as a common factor. So I can have GM. And then I've got one times or one over, then I've got two times, two times 10 to the six and that's in meters minus one over and then I've got five times 10 to the 10th. Um And that, that's, I'm gonna have to square root that, so I've got to square root that whole entire thing and that's gonna equal the final. So that's going to be, let's see, I've got square roots 6.67 times 10 of the minus 11. Then I've got the um mass of the planet which is seven times 10 to the 22. And then I've got, let's see, 1/4 times 10 to the sixth minus 1/5 times 10 to the 10th. And then you plug all of that stuff in just very carefully. You should get a final velocity is equal to 1.08 times 10 to the third meters per second. That is the velocity of both of these planets as they finally collide into each other. So let me know if you guys think about this um about the solution. Let me know if you guys have any questions.
4
Problem
Problem
You launch a rocket with an initial speed of m/s from Earth's surface. At what height above the Earth will it have ¼ of its initial launch speed? Assume the rocket's engines shut off after launch.
A
m
B
m
C
m
D
m
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